We develop a model that relates the polarization direction of a dipole sonic fast shear wave propagating along the borehole (hereafter called FSA) to the relative deviatoric stress tensor for arbitrary well orientations. We first define stress quantities, or “subsidiary principal stress,” relevant for sonic dipole shear characterized sufficiently far away into the formation to be unaffected by borehole stress concentration. Next, we show analytically for wells oriented within principal stress planes and numerically for wells outside principal stress planes that the stress-induced FSA coincides with the maximum normal stress direction orthogonal to the borehole (“maximum subsidiary principal stress”) using a nonlinear elastic model for isotropic unstressed background media and plane wave solutions. This model is independent of stress sensitivity parameters. This result is a consequence of, first, the direct relationship between the stressellipsoid factor and the shear stiffness difference ratio, [Formula: see text], and, second, the almost elliptical nature of stress-induced orthorhombic media. Using published laboratory data relating stiffnesses to applied differential stress for isotropic unstressed limestone and sandstone rocks, the model is confirmed for most well orientations except in the vicinity of the stress nodal point within the maximum and minimum principal stress plane, due to small anellipticity as stress increases. The orientations near the nodal point are also the zones of weakest stress-induced anisotropy zones and are therefore the least likely to be measured in practice as long as a quality indicator depending on both the degree of anellipticity and the strength of the shear anisotropy is below 0.5. Finally, we present a synthetic example of the application of this model to estimate the relative deviatoric stress tensor using two wells with different orientations.
Second-invariant-preserving Remap of the 2D deviatoric stress tensor in ALE methods
